Phase measuring method and apparatus for multi-frequency interferometry

ABSTRACT

The invention provides a novel method for absolute fringe order identification in multi-wavelength interferometry based on optimum selection of the wavelengths to be used. A theoretical model of the process is described which allows the process reliability to be quantified. The methodology produces a wavelength selection which is optimum with respect to the minimum number of wavelengths required to achieve a target dynamic measurement range. Conversely, the maximum dynamic range is produced from a given number of optimally selected wavelengths utilised in a sensor. The new concept introduced for optimum wavelength selection is scalable, i.e. from a three wavelength system to a four wavelength system, from four wavelengths to five, etc.

The present invention relates generally to phase measuring sensors formeasuring the phase of electromagnetic waves and, in particular, thoughnot exclusively, to methods for determining absolute fringe order infringe patterns obtained in such sensors.

The phase of electromagnetic waves projected onto an object can beaffected by various measurands of the object such as deformation [1],vibration [2], and refractive index variation due to density changes.Phase measuring sensors are often used for singe point or whole fieldprofilometry, for example profiling of three-dimensional objects. Thereexist a host of interferometric based techniques for single point andwhole field profilometry, from synthetic aperture radar (SAR) [3] tofringe projection [4]. In fringe projection techniques, a projectedwavelength is generated by projection of a known fringe pattern(sometimes referred to as a “fringe map”) onto the object ofinvestigation, under a certain angle of incidence. In interferometricbased sensors the third dimension of an object is coded in aninterferogram or (in fringe projection techniques) in a deformed fringepattern. In the following it will be understood that a deformed fringepattern can be analysed, and phase information extracted therefrom, inthe same way as an interferogram.

In an interference pattern there are two elements that contribute to thephase measurement dynamic range: the sub-fringe phase resolution and thenumber of fringes (fringe orders) spanned by the measurement. There istypically a simple function relating the measured optical fringe phaseto the desired measurand. The process of phase measuring producessub-fringe resolution, typically 1/100^(th) to 1/1000^(th) of a fringe.The sub-fringe resolution is calculated via either phase stepping [5] orFourier transform techniques [6]. However, the interferometric phase iscalculated using an inverse trigonometry function with principal valuesover the range −π to +π at best. Therefore the required phaseinformation is ‘wrapped’ into an interval with sharp discontinuities inthe data at the edges of that interval which must be spatially ortemporally unwrapped to obtain the fringe order information. Thisproblem is typically resolved by phase “unwrapping” (alternatively knownas fringe counting). Conventional interferometric analysis for singlepoint data relies on a temporal scan to measure the relative phasechange in going from one state to another. In the case of whole fielddata, a spatial unwrapping is achieved using an appropriate algorithmgiving relative information on the state (or change of state) across theimage field [7].

A big disadvantage of conventional phase measuring systems is that only“relative” information can be obtained. Many industrial measurementsrequire absolute information, for example in the measurement of range.In other cases absolute data is needed to overcome implementation issueswhich mean it is not possible to fringe count sufficiently rapidly totrack from one state to another and so an absolute measurement of eachstate is necessary. In whole field sensors, a generic problem common tomost interferometric techniques is the determination of absolute fringeorder in interferograms/deformed fringe patterns containing phasediscontinuities or spatially or temporally discrete samples. In suchcases it is difficult to unambiguously determine fringe orders—givingabsolute data—in the spatial unwrapping process. The spatial unwrappingprocess relies on the field being spatially contiguous i.e. it cannot beapplied to determine relative phases for spatially separate objects orobjects containing a discontinuity.

One approach to the unambiguous determination of fringe orders has beenthrough the use of two wavelength interferometry techniques [see H Zhao,W Chen, Y Tan, Phase unwrapping algorithm for the measurement of 3Dobject shapes, Applied Optics, 33, 4497-4500 (1994)]. Here twowavelengths are heterodyned to generate a beat wavelength. In twowavelength interferometry using wavelengths λ₀ and λ₁, with λ₀<λ₁, theunambiguous measurement range is increased (compared with singlewavelength interfrometry) to the synthetic wavelength Λ₀₁ at the beatfrequency, where Λ₀₁=λ₀λ₁/(λ₁−λ₀). From this, absolute data can beextracted. The ability to calculate a fringe order from measurements attwo wavelengths removes the need to spatially unwrap, assuming that theheterodyne process is robust. However, there is a finite, non-zero,phase measurement noise in two wavelength heterodyne systems and as thisphase noise increases the number of fringes which may be successfullyheterodyned decreases. This means that two wavelength heterodyne sensorstend not to be very robust, in terms of giving unambiguous absolutedata.

Another well established research area for absolute phase measurement isfringe projection for profilometry where the projected wavelengthsproduced can be varied relatively easily by varying the projected fringepattern or the angle of incidence. Therefore techniques based ontemporal phase unwrapping may be applied. Huntley et al introduced atemporal phase unwrapping approach using a reverse exponential series ofprojected fringes given by: s,s−1,s−2,s−4, . . . ,s/2 [8]. This is thesubject of International Patent Application Publication No.WO 97/36144.This technique allows 2^(N−1) fringes to be counted absolutely where Nis the number of fringe frequencies used. Here, the unwrapping isperformed between consecutive phase measurements to scale the fringeorder calculated at the subsequent wavelength. Absolute fringe order isobtained as the unwrapping is performed at each pixel independentlyalong the time axis. The number of projected fringe frequencies requiredis (log₂(s)+1). For each projected fringe frequency four phase steppedimages are obtained to determine the wrapped phase values in theinterval −π to +π. For example, to measure over a range of 32 fringes 6sets of 4 frames are needed, and for 128 fringes 8 sets of 4 frames.Therefore, considerable time is required to obtain the image frames, andas each image may contain >1 MB of information a significant dataprocessing problem is generated. Furthermore, a mechanism must exist togenerate the wide range of wavelengths required. This would be aparticular problem if a technique other than fringe projection werebeing used as then it would be the optical wavelength which must bevaried (not simply a synthetic wavelength).

It is an object of the present invention to avoid or minimise one ormore of the foregoing disadvantages.

According to a first aspect of the invention there is provided amulti-frequency interferometry method for measuring the absolute phaseof an electromagnetic wave, wherein the method comprises:

-   -   selecting a target measurement range, L, within which absolute        phase measurements are desired to be made, said phase        measurements being related to a desired measurand of an object;    -   determining a level of phase noise which will be present in        wrapped phase measurements which will be made;    -   for the selected target measurement range, and the determined        phase noise level, calculating an optimum number, N, of        wavelengths of electromagnetic radiation to be used in the        multi-frequency interferometry, where N≧3;    -   selecting an optimum series of values of said N wavelengths to        achieve optimum noise immunity in wrapped phase measurements to        be made; and    -   carrying out multi-frequency interferometry at the selected        values of the N wavelengths so as to make at least one wrapped        phase measurement at each of the N wavelengths, which wrapped        phase measurements are processed to obtain an absolute phase        measurement related to the desired measurand of the object.

The processing of the wrapped phase measurements may include one or moreof heterodyne processing, Fourier series processing, recursiveunwrapping and iterative unwrapping.

The optimum number, N, of wavelengths is preferably the minimum numberof wavelengths required to obtain unambiguous phase measurements in theselected target measurement range, for the determined phase noise level.The method may include proposing a measure of process reliabilityassociated with the determined phase noise level, corresponding to aknown probability of success in fringe order identification. Forexample, a process reliability of 6σ may be proposed, where σ is thestandard deviation noise in a function (for example in the form of afringe pattern) to be analysed: this function could, for example, be adiscrete level heterodyne function generated from the wrapped phasemeasurements made at two different wavelengths (this discrete levelheterodyne function will be used to obtain absolute phase data). Theproposed process reliability is preferably used to calculate the minimumnumber, N, of wavelengths required to obtain unambiguous phasemeasurements in the selected target measurement range.

The selected values of said N wavelengths may define a geometric seriesor, alternatively, may be combined in post-processing techniques, forexample heterodyne processing, to generate a geometric series ofsynthetic wavelengths. Where heterodyne processing is used, preferablythe selected values of the N wavelengths are chosen such that only oneheterodyne operation is required to generate each synthetic wavelengthin the geometric series. If more than one heterodyne operation isrequired to generate a desired synthetic wavelength it will beappreciated that measurement noise levels will increase. In a preferredembodiment, the values of said N wavelengths may be selected inaccordance with the following equation:1/λ₁=1/λ₀−(1/λ₀)^(i−1/N−1)(1/L)^(N−i/N−1)  (Equation A)where i=1, . . . ,N, where N is the number of wavelengths, λ₀ is thewavelength associated with the largest frequency, and λ_(i) is thewavelength of the ith frequency, and L is the target measurement range.However, there are also other possible series of projected wavelengthswhich will generate the desired geometric series of syntheticwavelengths.

The inventive method may be applied in many different types ofinterferometry, for example, in single point and full fieldprofilometry, synthetic-aperture radar (SAR) interferometry,multi-aperture synthesis techniques, and fringe projection techniques.For example, the method may be used for measuring the three-dimensionalshape of an object which may have surface discontinuities. In this case,the target measurement range is preferably the range over which(absolute) depth measurements relating to the profile of the object aredesired. (Alternatively, the target measurement range may be the desiredfield of view of the imaging system in which the method is applied.) Thestep of carrying out the multi-frequency interferometry may convenientlycomprise:

-   -   recording a series of fringe patterns obtained when the object        is illuminated with electromagnetic radiation at each of the        selected N optimum wavelengths;    -   processing the recorded fringe patterns so as to obtain wrapped        phase measurements in the form of a wrapped phase map for each        of the N optimum wavelengths, and processing the wrapped phase        maps to determine absolute fringe orders in the fringe patterns.        The absolute fringe orders may be calculated using heterodyne        processing and/or iterative unwrapping of the wrapped phase        maps. The determined absolute fringe orders can then be used to        unambiguously unwrap the wrapped phase maps, from which        unwrapped phase maps a three-dimensional profile of the object        can be compiled.

It will be appreciated that, depending on the particular interferometrytechnique being used, there will be various ways of illuminating theobject with radiation at the desired optimum wavelengths. For example,in fringe projection based interferometry, white light illumination isused to project a predetermined fringe pattern on to the object to bemeasured. In this case, it will be appreciated that the object is ineffect illuminated at a synthetic wavelength (hereinafter referred to asthe “projected wavelength”), this projected wavelength being determinedby the chosen fringe pattern projected onto the object. In the presentinvention, if the multi-frequency interferometry is carried out usingfringe projection, the selected series of N optimum wavelengths aretherefore different projected wavelengths at which the object must beilluminated. In other forms of interferometry the selected series of Noptimum wavelengths may be actual different wavelengths of light,normally in the form of laser beams, with which the object must beilluminated (usually sequentially). Thus, the recorded fringe patternsmay be interferograms (produced in phase measuring sensors, includinginterferometers and single point and full-field systems) or may bedeformed fringe patterns produced in fringe projection interferometry.

If the multi-frequency interferometry is carried out using fringeprojection, the optimum values of the projected wavelengths may beselected in accordance with the following re-written version of EquationA: $\begin{matrix}{{N_{f_{i}} = {N_{f_{0}} - \left( N_{f_{0}} \right)^{\frac{i - 1}{\lambda - 1}}}},{{{for}\quad i} = 1},2,\ldots\quad,\lambda} & \left( {{Equation}\quad B} \right)\end{matrix}$where λ is the number of projected wavelengths, N_(f) ₀ is the number ofprojected fringes in the largest fringe set, and N_(f) ₁ is the numberof projected fringes in the i^(th) fringe set. Alternatively, where thenumber, N, of wavelengths is three, the selected values of the threewavelengths may be N_(f) ₀ , N_(f) ₀ −√{square root over (N_(f) ₀ )},and √{square root over (N_(f) ₀ )}−1.

Preferably, at each said optimum wavelength a series of fringe patternsis recorded. Most preferably, a series of phase-stepped fringe patternsis recorded at each of the N wavelengths with which the object isilluminated. The phase-stepped fringe patterns can be used to create awrapped phase map for each said optimum wavelength. Alternatively, thewrapped phase maps may be obtained using Fourier transform processing ofa single fringe pattern recorded at each said optimum wavelength.

The object may be illuminated separately and sequentially with each ofthe selected optimum (real or projected) wavelengths of electromagneticradiation. Alternatively, the object may be illuminated with white lightand a plurality of the desired fringe patterns may be capturedsimultaneously by recording them with an image detector which cansimultaneously capture image data at the selected optimum wavelengths.For example, if the selected optimum wavelengths lie in discrete blue,green and red colour bands, a colour camera can be used to separatelybut simultaneously capture image data at red, green and bluefrequencies. In another possible embodiment, the inventive method may beused in high speed single point ranging applications. In this case thedesired measurand is of course absolute range. In one possibility, themulti-frequency interferometry may be carried out by illuminating theobject using a broadband femtosecond laser. Preferably, the output ofthe laser is split into two parts: one part illuminates the object andthe other part is used as a reference arm. The interference of theobject and reference light produces an interferogram in the from of atime-varying intensity signal which may conveniently be recorded using asingle point detector, for example a photodiode. The spectral range ofthe broadband pulses emitted by the laser is chosen so as to include thedesired optimum series of values of said N wavelengths, for the targetmeasurement range, L, selected in accordance with the inventive method.Other unwanted wavelengths can be filtered out, so that onlyinterferograms at each desired optimum wavelength are recorded. Phasemeasuring means may be provided to enable a wrapped phase measurement tobe obtained for every, or nearly every, femtosecond pulse emitted by thelaser. The absolute fringe orders in the interferograms may bedetermined by, for example, heterodyne processing or iterativeunwrapping of the wrapped phase measurements.

From the above it will be generally appreciated that the inventionprovides a novel strategy for absolute fringe order identification inmulti-wavelength interferometry based on optimum selection of thewavelengths to be used. As will be described in detail below withreference to the preferred embodiments, a theoretical model of theprocess has been developed which allows the process reliability to bequantified. The methodology produces a wavelength selection which isoptimum with respect to the minimum number of wavelengths required toachieve a target dynamic range. Conversely, the maximum dynamic range isproduced from a given number of optimally selected wavelengths utilisedin a sensor. Thus, according to a further aspect of the invention thereis provided a method of measuring absolute fringe order in a phasemeasuring sensor, wherein the method comprises:

-   -   selecting a number, N, of wavelengths of electromagnetic        radiation to be used to illuminate an object, where N≧3;    -   determining a level of phase noise which will be present in        wrapped phase measurements which will be made;    -   for the selected number, N, of wavelengths, and the determined        phase noise level, selecting optimum values of the N wavelengths        so as to achieve a maximum measurement range in which wrapped        phase measurements can be made relating to a desired measurand        of the object to be illuminated;    -   recording a series of fringe patterns obtained when the object        is illuminated with electromagnetic radiation at each of the        selected N optimum wavelengths; and    -   processing the recorded fringe patterns to obtain at least one        wrapped phase measurement at each of the selected N optimum        wavelengths, and processing the wrapped phase measurements to        determine absolute fringe orders in the fringe patterns.

According to another aspect of the invention there is provided amulti-frequency interferometer apparatus for shape measurement,comprising:

-   -   fringe projection means for generating a projected wavelength of        illumination by projecting a predetermined fringe pattern on to        an object; and    -   image capture and recording means for capturing and recording a        deformed fringe pattern obtained when said predetermined fringe        pattern is projected on to the object, the image capture means        being disposed at an angle to the direction of illumination of        the object with the projected fringe pattern; and    -   data processing means for processing recorded deformed fringe        patterns so as to obtain phase measurements therefrom; wherein    -   the fringe projection apparatus is variable such that an optimum        series of projected wavelengths may be generated, the values of        the wavelengths being such that, for a known level of phase        noise in the apparatus, and a chosen number of projected        wavelengths in said series, absolute fringe orders in the        deformed fringe patterns are measurable over a maximum        measurement range.

The fringe projection means may be any known means for producingprojection fringes. For example, the fringe projection means maycomprise a coherent fibre fringe projector for producing a pattern ofYoung's fringes across the object, or may comprise a spatial lightmodulator based fringe projector, or imaging means for imaging gratingpatterns on to the object.

It will be appreciated that the new concept introduced for optimumwavelength selection is readily scalable, i.e. from a three wavelengthsystem to a four wavelength system, from four wavelengths to five etc.Therefore, whilst the previous technology described in WO97/36144 andreference [8] allowed 2^(N−1) fringes to be counted absolutely where Nis the number of fringe frequencies used, the new technique scales asn^(N−1), where n is an arbitrary real number limited by phase noise of apractical interferometer.

Preferred embodiments of the invention will now be described by way ofexample only and with reference to the accompanying drawings in which:

FIG. 1 illustrates graphically the three discrete level heterodynefunctions for an optimum 4−λ process;

FIG. 2 illustrates graphically the three discrete level heterodynefunctions for an optimum 4−λ process with a total of 80.765 fringes;

FIG. 3 is a graph of Fringe Order against Sample Number across an areadetector, where the Sample Number is a function of position across anobject, calculated for the same optimum 4−λ process as FIG. 2, having atotal of 80.765 fringes;

FIG. 4(a) is a schematic diagram of an experimental set-up for fringeprojection shape measurement;

FIG. 4(b) is a schematic diagram illustrating a typical fibre fringeprojector for use in the experimental set-up of FIG. 4(a);

FIG. 5 is an experimentally obtained fringe order map for optimum 3−λheterodyne processing; and

FIG. 6 is a schematic diagram of a high speed single point rangingsensor.

A detailed analysis of the theoretical basis of the invention will nowbe described, followed by a description of various practicalembodiments. This analysis is presented with reference to fringeprojection interferometry, but it will be appreciated that the analysisis equally applicable to all forms of interferometry in which it isdesired to determine absolute fringe order in an interferogram.

Theoretical Development

Two Wavelength Interferometry

In two wavelength heterodyne fringe projection interferometry toeliminate step height ambiguity the difference in the number of fringesprojected across the field of view must be <1. Let the number ofprojected fringes across the field be N_(λ1) for the is wavelength; thenN_(λ1)−N_(λ2)<1 with N_(λ1)>N_(λ2). The difference in wrapped phasescalculated at the two wavelengths, a heterodyne function, can beexpressed as a phase within the interval −π to π and will consist of amonotonic ramp across the image. A convenient representation is tocalculate a discrete phase level for each of the fringe orderscorresponding to N_(λ1). This is obtained by subtracting a scaledversion of the wrapped phase at N_(λ1) from the heterodyne function—thescaling factor being given by (N_(λ1)−N_(λ2))/N_(λ1). In practice, sucha discrete level heterodyne function can only identify a limited numberof fringe orders owing to the presence of phase noise. Each N_(λ1) isknown, as it is set in a white light system or can be measured incoherent systems [9]. Each wrapped phase measurement contains phasenoise, which is modelled as a Gaussian distribution with zero mean and astandard deviation of σ_(φ) [10]. The heterodyne function then containsnoise with a standard deviation given by √{square root over (2)}π_(φ).We define a process robustness of 6σ, corresponding to a probability of99.73% success in fringe order identification within the measurementsystem. Therefore, the discrete phase levels must be separated in phaseby at least 6√{square root over (2)}σ_(φ). Hence, for 6σ reliability thenumber of fringes which can be correctly identified is limited by:$N_{\lambda\quad 1} \leq {\frac{2\pi}{6\sqrt{2}\quad\sigma_{\phi}}.}$Defining the measurement dynamic range as the product of the phaseresolution and the number of fringes successfully numbered. Therefore,for a phase resolution of 1/100^(th) of a fringe, 0.12 fringes can benumbered to 6σ reliability, giving a dynamic range of 1200, which isinsufficient for most engineering applications.Optimally Selected Wavelengths in N Wavelength Interferometry

The introduction of a third projected wavelength allows the generationof two independent heterodyne functions containing N_(DL1) and N_(DL2)discrete levels, where N_(DL1)×N_(DL2)=N_(λ1). The discrete phase levelsin each modified heterodyne function will be separated by:∂_(DL1)=2π/N_(DL2), and ∂_(DL2)=2π/N_(DL1).  (1)In this arrangement it is found that as ∂_(DL1) increases ∂_(DL2)decreases. Therefore, for maximum overall reliability in fringe orderidentification the optimum set of projected fringe wavelengths is givenby the symmetrical arrangement where ∂_(DL1)=∂_(DL2) andN_(DL1)=N_(DL2)=√{square root over (N_(λ1))}. If the number of discretelevels is not balanced the effect is to increase one of N_(DL1) andN_(DL2), thereby bringing ∂_(DL1) or ∂_(DL2) closer to the noise limit.Hence, from equation (1), with an optimised three wavelength approachthe condition for 6σ reliability fringe order identification is givenby, $\begin{matrix}{\sqrt{N_{\lambda\quad 1}} \leq {\frac{2\pi}{6\sqrt{2\quad}\sigma_{\phi}}.}} & (2)\end{matrix}$The number of fringes which may be heterodyned reliably in an optimumthree wavelength heterodyne set-up is therefore the square of that forthe two wavelength case for the same phase measurement noise.

Equations 1,2 are used to define general expressions relating thenumbers of projected fringes at the N wavelengths. By assuming N_(f) ₀>N_(f) ₁ andN _(DL1) ×N _(DL2) × . . . ×N _(DL2) =N _(f) ₀ ,with N_(DL1)=N_(DL2)= . . . =N_(DL1), for optimum noise immunity, it ishere proposed that the numbers of projected fringes should be selectedin accordance with the following expression: $\begin{matrix}{{N_{f_{i}} = {N_{f_{0}} - \left( N_{f_{0}} \right)^{\frac{i - 1}{\lambda - 1}}}},{{{for}\quad i} = 1},2,\ldots\quad,\lambda} & (3)\end{matrix}$where: λ number of wavelengths, N_(f) ₀ number of fringes in the largestfringe set, N_(f) ₁ number of fringes in the i^(th) fringe set. The termN_(f) _(λ) =0 is included in order to generalize the expressions thatfollow. The general formulas to calculate the fringe order are given by:$\begin{matrix}{{DL}_{i} = {H_{0,i} - {\frac{N_{f_{0}} - N_{f_{i}}}{N_{f_{0}} - N_{f_{i + 1}}} \times H_{0,{i + 1}}}}} & (4) \\{{IDL}_{i + 1} = {{NINT}\left\lbrack {\left( {{DL}_{1} + {2\quad\pi \times {IDL}_{1}}} \right) \times \left( \frac{N_{f_{0}} - N_{f_{i + 1}}}{N_{f_{0}} - N_{f_{i}}} \right) \times 2\quad\pi} \right\rbrack}} & (5)\end{matrix}$for i=1, . . . , λ−1, where DL_(i) is the i^(th) discrete levelfunction, IDL_(i) is the is integer discrete level, H_(0,i) is theheterodyne between the 0 and i^(th) wrapped phase map and NINT denotestaking the nearest integer. The recursive relationship for the integerdiscrete level functions is initialised by setting IDL₀=0. The fringeorder for the wrapped phase map with N_(f) ₀ fringes is given by IDL_(λ)from equation (5).

For example using four synthetic (i.e. projected) wavelengths with N_(f)₀ =64, then N_(f1)=63, N_(f2)=60, N_(f3)=48, FIG. 1 shows the discretelevel functions resulting from a simulation of the process. The phasedifference between discrete levels (y-axis) is equal in each plot asexpected for an optimum wavelength configuration. In this case N_(λ1)=64and this means the discrete level functions contain 1 fringe with 4discrete levels on it (1*4=4 fringe orders), 4 fringes each with 4discrete levels on them (4*4=16 fringe orders), and 16 fringes each with4 discrete levels on them (4*16=64 fringe orders). So we have thepattern 1, 4, 16, 64, or to be generic: (N_(λ1))^(1/3), (N_(λ1))^(2/3),N_(λ1), where we note that we are dealing with cube roots which is thenumber of wavelengths-1. The number sequence here defines a geometricseries, where the factor relating neighbouring terms is N_(DL). Thegeometric series can be expressed as$\left( N_{f_{0}} \right)^{\frac{i - 1}{\lambda - 1}},{{{for}\quad i} = 1},2,\ldots\quad,\lambda,$where λ is number of wavelengths.

A multiplicative intensity noise of 2.5% has been applied to the cosineintensity fringes which is representative of scientific CCD cameras andis a primary source of phase noise.

It is here proposed that the number of fringes that may be numbered to6σ reliability for a system with λ wavelengths is given by (fromequation (2) above): $\begin{matrix}{\sqrt[{\lambda - 1}]{N_{\lambda\quad 1}} \leq \frac{2\pi}{6\sqrt{2\quad}\sigma_{\phi}}} & (6)\end{matrix}$The reliability of the process only depends on the first and lastwrapped phase maps with all intermediate values for other wavelengthscancelling out (see equations 4 and 5). As the last wavelengthcorresponds to zero fringes, rather than performing a phase measurementhere it is more accurate to set the wrapped phases for this wavelengthto zero. Hence the process reliability depends only on the error in thewrapped phase map at N_(f) ₀ . Therefore, as the number of wavelengthsused is increased the process reliability remains unchanged and thenumber of fringes that may be labelled increases as given by equation 6.

An alternative representation of equation 6 is useful for someapplications and obtained by noting that L/λ_(λi)=N_(fi) where: λ_(λi)is the wavelength associated with the i^(th) frequency and L is thedesired unambiguous measurement range. Equation 6 would then bere-expressed as: $\begin{matrix}{\sqrt[{\lambda - 1}]{\frac{L}{\lambda_{\lambda\quad 1}}} \leq {\frac{2\pi}{6\sqrt{2}\sigma_{\phi}}.}} & (7)\end{matrix}$FIG. 1 shows simulated (i.e. computer generated) phases calculated atthe synthetic (i.e. projected) wavelengths for optimum four-frequencyinterferometry with L/λ₀=64.

In any measurement situation the phase noise will be measured and henceknown, and the target measurement range, L, will be known. Then equation6 can be used to determine the number of wavelengths needed to makeN_(λ1) large enough to span the desired measurement range, in that:N_(λ1)=L/_(λ) ₀, where λ₀ is the wavelength associated with the largestfrequency. The result of this will be the minimum number of wavelengths,λ, needed to achieve the target measurement range for the particularsensor system.

It should be noted that in the example of FIG. 1, the value (N_(f) ₀)^(λ−1) is an integer and therefore the discrete levels are equal inphase values for each discrete level function. For any real positivenumber, (N_(f) ₀ )^(λ−1), the algorithms function correctly. Forexample, N_(f) ₀ =80.765 and using four projected fringe wavelengths thediscrete level functions are given in FIG. 2 (again showing simulatedphases). The recursive unwrapping process correctly unwraps successivebeat fringes to obtain the fringe order, as seen in FIG. 3.

For a typical four wavelength system σ_(φ)=2π/85 and the number offringes that can be correctly identified with 6σ reliability is >1000,giving a dynamic range of >85000. The following table (Table 1) showsvalues for N_(f) ₀ and dynamic range for a range of λ and σ_(φ). TABLE 1Numbers of fringes and dynamic range obtained with 99.73% reliabilityfor optimum 3-λ and 4-λ heterodyne processing. Phase Optimum OptimumNoise 3-λ Heterodyne 4-λ Heterodyne (rms, Dynamic Dynamic radians)N_(∫0) Range N_(∫0) Range 2π/50 35 1,736 205 10,230 2π/100 139 13,8891,637 163,700 2π/200 556 111,111 13,095 2,619,000 2π/400 2,222 888,889104,707 41,902,800It will be understood that although in the above description we haveproposed 6σ process reliability, the process reliability may be definedby other values as desired, for example 8σ reliability, in which caseequation (6) becomes:$\sqrt[{\lambda - 1}]{N_{\lambda\quad 1}} \leq \frac{2\pi}{8\sqrt{2}\sigma_{\phi}}$Alternative Strategies for Generating Optimum WavelengthsThere are alternative ways to calculate the numbers of fringes used ateach wavelength that result in optimum configurations in the discretelevel functions. For example using the three projected wavelengths givenby N_(f) ₀ , N_(f) ₀ −√{square root over (N_(f) ₀ )}, √{square root over(N_(f) ₀ )}1, these can be heterodyned together as follows to obtain thedesired geometric series 1, √N_(f) ₀ , N_(f) ₀ : heterodyne N_(f) ₀ withN_(f) ₀ −√{square root over (N_(f) ₀ )} to get √N_(f) ₀ , heterodynethis with √{square root over (N_(f) ₀ )}−1 to get 1. Equations 4 and 5may than be applied as before. It is expected that the noise floor isworse in this case (than for wavelengths selected in accordance withEquation 3) as it relies on two heterodyne operations to get one of thesynthetic wavelengths (the 1) in the geometric series. There are alsomany other possibilities to arrive at the desired geometric series 1,√N_(f) ₀ , N_(f) ₀ , the only difference in the various ways is thenumber of heterodyne operations required in each case. However, thenoise level will increase the greater the number of heterodyneoperations that are required to obtain any synthetic wavelength in thedesired geometric series.

In fact, if the maximum number of heterodyne operations to get to amember of the geometric series is denoted by r, the measurement rangeequation 6 can be re-expressed as: $\begin{matrix}{\sqrt[{\lambda - 1}]{N_{\lambda\quad 1}} \leq \frac{2\pi}{6\sqrt{{r + 1}\quad}\sigma_{\phi}}} & (7)\end{matrix}$For example, in the three wavelength case, if the original projectedwavelength selection is 100, 99, 90 a single heterodyne is needed to getto each term in the desired geometric series 1,10,100 and hence r=1,leading to equation 6.

In fact, in the case of fringe projection using three wavelengths, onecould capture directly a fringe pattern with the projected wavelengthscorresponding to 1, √N_(f) ₀ , and N_(f) ₀ fringes and apply iterativeunwrapping to these directly. This would be instead of the more usualcapture of N_(f) ₀ , N_(f) ₀ −1, N_(f) ₀ −√N_(f) ₀ (as proposed byequation 3), then heterodyne (i.e. “beat”) N_(f) ₀ with N_(f) ₀ −1 toget 1, and beat N_(f) ₀ with N_(f) ₀ −√N_(f) ₀ to get the √N_(f) ₀ andthen apply iterative unwrapping to the derived data at 1, √N_(f) ₀ , andN_(f) ₀ fringes.

Capturing directly a fringe pattern at the series 1, √N_(f) ₀ and N_(f)₀ fringes thus corresponds to the case r=0 in equation 7 above. In thiscase it will be appreciated that discrete level functions can be formeddirectly in the processing stage without heterodyning first. Thediscrete level function is obtained by subtracting a scaled version ofthe wrapped phase at the next higher frequency in the series, e.g. for 3wavelengths to get √N_(f) ₀ levels on the single unambiguous fringe thephase measured with √N_(f) ₀ fringes is multiplied by 1/√N_(f) ₀ andsubtracted from the phase obtained with a single fringe. The discretelevel functions may then be iteratively unwrapped. The geometric series1, √N_(f) ₀ , N_(f) ₀ still provides an equal factor of √N_(f) ₀ ingoing from 1 to √N_(f) ₀ to N_(f) ₀ . As these factors are equal, whenthe discrete level functions are calculated they will contain equalnumbers of discrete levels per fringe and hence the process reliabilityis equal in the discrete level functions and the reliability of theprocess overall is optimised. From this it will be understood that insome possible embodiments of the invention the projected wavelengths maybe chosen so as to match exactly the desired geometric series i.e. theseries defined by$\left( N_{f_{0}} \right)^{\frac{i - 1}{\lambda - 1}},{{{for}\quad i} = 1},2,\ldots\quad,\lambda,$where λ is number of wavelengths.Modifications to the Theoretical Description

To allow for experimental error in obtaining exactly the number offringes required across the measurement range, the wavelengths would beselected close to their theoretical values but in such a way as todefine an unambiguous measurement range slightly larger than thatrequired. For example, the second wavelength (i=2, in equation 3) may bemodified by up to 30% in order to guarantee forming less than one beatfringe across the desired measurement range. Therefore, in practicalsystems it would be common to use N_(f) ₀ , N_(f) ₀ −0.7, N_(f) ₀−√N_(f) ₀ for a three wavelength system.

Experimental Demonstration

A whole field, triangulation based shape measurement system is given asthe exemplar for the above theoretical analysis. In this system acoherent fibre fringe projector 10 is used to produce a pattern ofYoung's fringes 8 across a test object 12. A CCD camera 14 is thenplaced at an angle θ to the illumination direction to view the fringes,as shown in FIG. 4(a). The CCD camera 14 is connected to a computersystem 16 for implementing the data processing scheme (in particular,the heterodyne operations and the recursive unwrapping algorithmsspecified by equations (4) and (5) above) and control of the dataacquisition process. The computer system includes a memory for recordingfringe patterns captured by the CCD camera 14. The computer system isprogrammed to carry out the phase unwrapping and fringe orderidentification processes as herein described in accordance with theinvention. Increasing the number of projected fringes or the angularseparation θ between the CCD camera 14 and the fringe projector 10increases the sensitivity to the depth (z) of the test object 12. FIG.4(b) illustrates in more detail the features of the fibre fringeprojector 10. Like parts in FIGS. 4(a) and (b) are referenced by likenumerals. In known manner, this comprises two output fibres 20,21 havingtheir ends placed side-by-side with a spacing therebetween. One of theoutput fibres 20 was wrapped around a cylindrical PZT 24 to allow theoptical phase to be modulated. A servo control system 26 allows thephase of the projected fringes to be stabilised by monitoring thereflections from the distal ends of the output fibres at the 4^(th) armof a directional coupler 30. The servo also enables accurate 90° phasesteps to be obtained [11]. A second CCD camera 32 was incorporated intothe fringe projector to measure the fibre separation, and hence thenumber of projected fringes. The fringes were sampled directly onto thesecond CCD camera 32 via a polarising beamsplitter 34. A measurement ofthe phase distribution across this CCD camera 32 gives the fibreseparation and hence the number of projected fringes [9]. The fibreseparation can be varied by means of a linear traverse (not shown) onwhich one of the fibre ends is mounted. We have demonstrated aresolution of <10 nm in the fibre separation measurement compared to a50 nm requirement predicted by a simulation for three projectedwavelengths. To demonstrate the process, the flat side of an object wasassessed initially as this gave easy verification of the fringe ordercalculation. Three wavelengths were used (from equation 3 above) with100, 99 and 90 projected fringes. FIG. 5 shows the result of applyingequations 4 and 5 to the three wrapped phase maps obtained. The centralfringe is automatically identified by the heterodyne process and iscoloured black. The fringe orders are clearly identified as a repeatingscheme of 6 colours either side of the central fringe.

The phase resolution obtained was estimated to be 1/80^(th) fringeprincipally limited by the random speckle phase which contributes to theinterference phase. With σ_(φ)=2π/80 and N_(f0)=100 equation 2 can bemanipulated to determine the fringe order numbering reliability expectedfrom theory which evaluates to 99.53%. A local neighbourhood checkapplied to the central flat area of the experimental data in FIG. 5shows the number of pixels giving the correct fringe order to be 99.52%.Therefore, a validation of the process and the underlying theory hasbeen demonstrated.

The new process is equally applicable to white light fringe projectorsbased on spatial light modulators or on custom gratings produced onglass substrates. Compared to the reverse exponential temporalunwrapping scheme [8], to measure over ˜100 fringes requires 8 wrappedphase maps to be obtained. Therefore, the use of the new algorithmenables a reduction in data acquisition time and data space of >60%. Thedynamic range obtained in this measurement was 1:8000.

The above described embodiment provides a robust measurement processbased on multiple wavelengths, phase measurement and heterodyneprocessing. It is shown that given a certain phase measurement noise itis possible to maximise the number of fringe orders unambiguouslyidentifiable and therefore the measurement dynamic range. Each fringeorder can be identified as an absolute number. A robustness measurebased on Gaussian statistics is derived for the process.

It will be appreciated that the inventive process is applicable to anyform of interferometric sensor where phase measurements are available atdefined sensitivities (wavelengths), not simply only to fringeprojection based phase measuring sensors. For interferometric sensorsnot based on fringe projection, the equivalent of the optimum noiseimmunity condition N_(DL1)=N_(DL2)= . . . =N_(DLi) proposed above forfringe projection sensors is:Λ₀₁/Λ₀₂=Λ₀₂/Λ₀₃= . . . =Λ_(0, N−1)/λ₀  (8)

-   -   where Λ₀₁, Λ₀₂, Λ₀₃, . . . , Λ_(0, N−1) are the synthetic (beat)        wavelengths formed when λ₀ is heterodyned with λ₁, λ₂, λ₃, . . .        , λ_(N−1) respectively, e.g. Λ₀ is the phase difference between        phase measurements at λ₀ and λ₁ and is defined by:        Λ₀₁=λ₀λ₁/(λ₁−λ₀). In this case the equation (3) is re-expressed        as:        1/λ₁=1/λ₀−(1/λ₀)^(i−1/N−1)(1/L)^(N−i/N−1)  (Equation A)        where i=1, . . . ,N, where N is the number of wavelengths, λ₀ is        the wavelength associated with the largest frequency, and λ_(i)        is the wavelength of the ith frequency, and L is the target        measurement range. However, again it will be appreciated that        alternative series of wavelength values are possible which can        be combined in post-processing to produce the desired geometric        series of synthetic wavelengths satisfying equation 8 above.        With reference to equation (7) above, the r=0 case is also        possible (i.e. the wavelengths are chosen to match the desired        geometric series, without processing heterodyne operations being        required) as long as the required range of wavelengths is        available from the wavelength sources being used and are        detectable using sensors. In practice, the visible wavelength        range will not be broad enough to offer an r=0 solution, but an        example where it would be possible is in radar systems where the        wavelength range available is orders of magnitude broader.

Examples of alternative inventive embodiments (not based on fringeprojection) are:

-   -   a) High speed, single point ranging. In this application the aim        is to monitor one point on a target object at high speed. Short        pulse, pico and femtosecond, lasers produce a pulse of light        over a broad range of wavelengths (as the pulse length shortens        the wavelength range increases). FIG. 6 illustrates an example        embodiment. The light from a pico or femtosecond laser 40 is        split into two parts (using a first directional coupler 41), one        part goes to the object 50 and the other acts as a reference        beam 42 in a variable delay line 44 (for path length matching).        Light scattered by the object 50 is then combined with the        reference light (using a second directional coupler-43). For        each wavelength required, a filter 46 is used to isolate that        wavelength from the broad bandwidth pulse, with the filtered        light detected on a single point detector 48(a suitable example        would be a photodiode). The output from the detector 48 is        passed to a high speed data logger (not shown). In practise, a        pair of detectors would be used in quadrature for each        wavelength channel in order to calculate the optical phase.        Thereby a phase measurement is obtained on each pulse from the        laser. So the ‘interferograms’ (equivalent in purpose to the        deformed fringe patterns used in the above-described fringe        projection profilometry embodiment of FIG. 4) are mapped out in        time from each detector as an intensity signal. The absolute        phase information is obtained by post-processing the        interferograms obtained at each wavelength in a similar manner        to the post-processing of the wrapped phase maps obtained in the        FIG. 4 embodiment, for example using heterodyne processing to        generate data at the desired optimum geometric series of        synthetic wavelengths, to extract the absolute phase data (from        which absolute range measurements are calculable).    -   b) Colour camera application. In this embodiment simultaneous        illumination of the object occurs at three wavelengths. The        three wavelengths are chosen to reside in discrete colour bands,        e.g. the red, green and blue components of a colour camera        (having a known arrangement of red, green and blue pixels) with        zero or minimal leakage of one component into the neighbouring        colour band A colour camera provides simultaneous acquisition of        the images in each of the colour bands. The colours may simply        be used as a means of separating out the multi-wavelength image        obtained at the colour camera in which case it would be        preferable to obtain directly the data at the 1, √N, and N        fringes, where N is the number of fringes in the highest        resolution fringe set. Alternatively, the wavelengths would be        selected to give (with some processing e.g. heterodyne        operations) the geometric series defined by 1, √N, and N        fringes. Phase data captured at these wavelengths could be        heterodyned to get to the 1, √N, and N fringes and hence        iteratively unwrapped. This latter approach may also be applied        to other forms of interferometric sensing. Therefore, the        multi-wavelength illumination would occur simultaneously. To        obtain the phase data there are a number of options the most        common of which is phase stepping. Here the phase of each fringe        pattern at each wavelength needs to be varied to give at least        three image sets and hence some temporal acquisition is still        required. The next most common process is Fourier transform        fringe analysis which derives the phase data from a single        image. It will be appreciated that using the latter technique        would enable all the required phase data to be captured in a        genuinely “single snapshot” technique.

REFERENCES

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1. A multi-frequency interferometry method for measuring the absolutephase of an electromagnetic wave, wherein the method comprises:selecting a target measurement range, L, within which absolute phasemeasurements are desired to be made, said phase measurements beingrelated to a desired measurand of an object; determining a level ofphase noise which will be present in wrapped phase measurements whichwill be made; for the selected target measurement range, and thedetermined phase noise level, calculating an optimum number, N, ofwavelengths of electromagnetic radiation to be used in themulti-frequency interferometry, where N≧3; selecting an optimum seriesof values of said N wavelengths to achieve optimum noise immunity inwrapped phase measurements to be made; and carrying out multi-frequencyinterferometry at the selected values of the N wavelengths so as to makeat least one wrapped phase measurement at each of the N wavelengths,which wrapped phase measurements are processed to obtain an absolutephase measurement related to the desired measurand of the object.
 2. Themethod according to claim 1, wherein the optimum number, N, ofwavelengths is the minimum number of wavelengths required to obtainunambiguous phase measurements in the selected target measurement range,for the determined phase noise level.
 3. The method according to claim 1wherein the method includes proposing a measure of process reliabilityassociated with the determined phase noise level, corresponding to aknown probability of success in fringe order identification.
 4. Themethod according to claim 3, wherein the proposed process reliability isσs, where σ is the standard deviation noise in a discrete levelheterodyne function generated from the wrapped phase measurements madeat two different wavelengths.
 5. The method according to claim 3 whereinthe proposed process reliability is used to calculate the minimumnumber, N, of wavelengths required to obtain unambiguous phasemeasurements in the selected target measurement range.
 6. The methodaccording to claim 1, wherein the processing of the wrapped phasemeasurements includes heterodyne processing.
 7. The method according toclaim 6, wherein the heterodyne processing produces a plurality ofdiscrete level heterodyne functions each containing equal numbers oflevels in the interval −π to +π.
 8. The method according claim 1,wherein the processing of the wrapped phase measurements includesiterative unwrapping.
 9. The method according to claim 1, wherein theselected values of said N wavelengths define a geometric series.
 10. Themethod according to claim 1, wherein the selected values of said Nwavelengths are such that they can be combined in a predetermined mannerto generate a geometric series of synthetic wavelengths.
 11. The methodaccording to claim 10, wherein the selected values of said N wavelengthsare heterodyne processed to generate the geometric series of syntheticwavelengths.
 12. The method according to claim 11, wherein the selectedvalues of the N wavelengths are chosen such that only one heterodyneoperation is required to generate each synthetic wavelength in thegeometric series.
 13. The method according to claim 10, wherein thevalues of said N wavelengths are selected in accordance with thefollowing equation:1/λ₁=1/λ₀−(1/λ₀)^(i−1/N−1)(1/L)^(N−i/N−1) where i=1, . . . ,N, where Nis the number of wavelengths, λ₀ is the wavelength associated with thelargest frequency, and λ_(i) is the wavelength of the ith frequency, andL is the target measurement range.
 14. The method according to claim 1,wherein the method is used for measuring the three-dimensional shape ofan object which may have surface discontinuities, and the targetmeasurement range is the range over which absolute depth measurementsrelating to the profile of the object are desired.
 15. The methodaccording to claim 14, wherein the step of carrying out themulti-frequency interferometry comprises: recording a series of fringepatterns obtained when the object is illuminated with electromagneticradiation at each of the selected N optimum wavelengths; processing therecorded fringe patterns so as to obtain wrapped phase measurements inthe form of a wrapped phase map for each of the N optimum wavelengths,and processing the wrapped phase maps to determine absolute fringeorders in the fringe patterns.
 16. The method according to claim 15,wherein the absolute fringe orders are calculated using at least one ofheterodyne processing, recursive unwrapping and iterative unwrapping ofthe wrapped phase maps.
 17. The method according to claim 15 wherein ateach said optimum wavelength a series of phase-stepped fringe patternsis recorded.
 18. The method according to claim 1, wherein themulti-frequency interferometry is carried out using fringe projection,and the selected series of N optimum wavelengths are syntheticwavelengths projected on to the object.
 19. The method according toclaim 18, wherein the selected values of the N synthetic wavelengthsprojected on to the object define a geometric series given by$\left( N_{f_{0}} \right)^{\frac{i - 1}{\lambda - 1}},{{{for}\quad i} = 1},2,\ldots\quad,\lambda,$where λ is number of wavelengths and N_(f) ₀ is the number of projectedfringes in the largest fringe set.
 20. The method according to claim 18,wherein the values of the N synthetic wavelengths projected on to theobject are selected in accordance with the following equation:${N_{f_{1}} = {N_{f_{0}} - \left( N_{f_{0}} \right)^{\frac{i - 1}{\lambda - 1}}}},{{{for}\quad i} = 1},2,\ldots\quad,\lambda$where λ is the number (1=N) of projected wavelengths, N_(f) ₀ is thenumber of projected fringes in the largest fringe set, and N_(f) ₁ isthe number of projected fringes in the i^(th) fringe set.
 21. The methodaccording to claim 18, wherein the number, N, of wavelengths is equal tothree, and the selected values of the three wavelengths are N_(f) ₀ ,N_(f) ₀ −√{square root over (N_(f) ₀ )}, and √{square root over (N_(f) ₀)}−1, where N_(f) ₀ , is the number of projected fringes in the largestfringe set.
 22. The method according to claim 1, wherein the object isilluminated separately and sequentially with each of the selectedoptimum wavelengths of electromagnetic radiation.
 23. The methodaccording to claim 14, wherein the object is illuminated with whitelight and a plurality of said fringe patterns are capturedsimultaneously by recording them with an image detector which cansimultaneously capture image data at the selected optimum wavelengths.24. The method according to claim 23 wherein the image detector is acolour camera.
 25. The method according to claim 1, wherein the methodis carried out in a single point ranging system in which the absolutephase measurements made are used to calculate absolute range.
 26. Themethod according to claim 25, wherein the multi-frequency interferometryis carried out by illuminating the object using a broadband femtosecondlaser.
 27. The method according to claim 26, wherein the spectral rangeof the broadband pulses emitted by the laser is chosen so as to includethe said optimum series of values of said N wavelengths.
 28. A method ofmeasuring absolute fringe order in a phase measuring sensor, wherein themethod comprises: selecting a number, N, of wavelengths ofelectromagnetic radiation to be used to illuminate an object, where N≧3;determining a level of phase noise which will be present in wrappedphase measurements which will be made; for the selected number, N, ofwavelengths, and the determined phase noise level, selecting optimumvalues of the N wavelengths so as to achieve a maximum measurement rangein which wrapped phase measurements can be made relating to a desiredmeasurand of the object to be illuminated; recording a series of fringepatterns obtained when the object is illuminated with electromagneticradiation at each of the selected N optimum wavelengths; and processingthe recorded fringe patterns to obtain at least one wrapped phasemeasurement at each of the selected N optimum wavelengths, andprocessing the wrapped phase measurements to determine absolute fringeorders in the fringe patterns.
 29. A multi-frequency interferometerapparatus for shape measurement, comprising: fringe projection means forgenerating a projected wavelength of illumination by projecting apredetermined fringe pattern on to an object; and image capture andrecording means for capturing and recording a deformed fringe patternobtained when said predetermined fringe pattern is projected on to theobject, the image capture means being disposed at an angle to thedirection of illumination of the object with the projected fringepattern; and data processing means for processing recorded deformedfringe patterns so as to obtain phase measurements therefrom; whereinthe fringe projection apparatus is variable such that an optimum seriesof projected wavelengths may be generated, the values of the wavelengthsbeing such that, for a known level of phase noise in the apparatus, anda chosen number of projected wavelengths in said series, absolute fringeorders in the deformed fringe patterns are measurable over a maximummeasurement range.
 30. The apparatus according to claim 29, wherein thefringe projection means comprises a coherent fibre fringe projector forproducing a pattern of Young's fringes across the object.